On Sloane's persistence problem

We investigate the so-called persistence problem of Sloane, exploiting connections with the dynamics of circle maps and the ergodic theory of $\mathbb{Z}^d$ actions. We also formulate a conjecture concerning the asymptotic distribution of digits in long products of finitely many primes whose truth would, in particular, solve the persistence problem. The heuristics that we propose to complement our numerical studies can be thought in terms of a simple model in statistical mechanics.Comment: 5 figure

Renyi Entropy Rate of Stationary Ergodic Processes

In this paper, we examine the Renyi entropy rate of stationary ergodic processes. For a special class of stationary ergodic processes, we prove that the Renyi entropy rate always exists and can be polynomially approximated by its defining sequence; moreover, using the Markov approximation method, we show that the Renyi entropy rate can be exponentially approximated by that of the Markov approximating sequence, as the Markov order goes to infinity. For the general case, by constructing a counterexample, we disprove the conjecture that the Renyi entropy rate of a general stationary ergodic process always converges to its Shannon entropy rate as {\alpha} goes to 1

Stochastic-like behavior in arithmetic dynamical systems : an investigation of collatz map hailstone sequences

Orientador: Prof. Dr. Marcos Gomes E. da Luz.Coorientador: Prof Dr. Madras Viswanathan GandhiDissertação (mestrado) - Universidade Federal do Paraná, Setor de Ciências Exatas, Programa de Pós-Graduação em Física. Defesa : Curitiba, 28/02/2023Inclui referências: p. 87-96Resumo: Dinâmica Aritmética é um a área de pesquisa emergente, que estuda o comportamento de sistemas em espaços e tempos discretos. A presente dissertação lida com um sistema dinâmico aritmético chamado Mapa de Collatz, uma regra para inteiros positivos n, tais que n -> n/2 para n par, e n -> 3n + 1 para n ímpar. Um renomado e não resolvido problema matemático conjectura que para qualquer inteiro positivo n, finitas iterações do Mapa de Collatz eventualmente atingirão 1. A sequência de inteiros da iteração do Mapa de Collatz a partir de uma condição inicial n0, até o ponto em que atinge 1, é chamada sequência de granizo. O Mapa de Collatz, apesar de fornecer um a dinâmica muito rica para números naturais, só começou a ser explorado recentemente no contexto de modelos e fenômenos físicos. Este trabalho descreve investigações na tentativa de caracterizar se as sequências de granizo podem ser vistas como um sistema determinístico realizando com portamento do tipo estocástico, buscando iluminar o caminho entre teoria de números e mecânica estatística, através da área de sistemas dinâmicos aritméticos estocásticos. Para fazer isso, análises estatísticas apropriadas em várias sequências foram feitas, utilizando um numeroso conjunto de condições iniciais muito grandes (até a ordem de n0 ~ 2^10000). O processo de amostragem de condições iniciais foi conduzido utilizando uma nova representação para inteiros positivos, com conexão direta com 2-ádicas, chamados vetores-m. Ao aplicar métodos de análise de séries temporais tais como Power Spectrum e Detrended Fluctuation Analysis, o comportamento do tipo estocástico é confirmado, reforçando a literatura acerca das sequências de granizo performarem Movimento Browniano Geométrico (MBG). Análises de função de autocorrelação e entropia de von Neumann mostram desvios em relação ao MBG para condições iniciais especiais, indicando fontes de determinismo e previsibilidade dentro da tendência geral estocástica das séries. Estes desvios aparecem na forma de autocorrelações de curto e médio alcance, bem como uma diminuição no valor das entropias para órbitas a partir destas condições iniciais. A entropia de von Neuman também permite a caracterização da estrutura interna da sequência, por meio da análise das componentes dos vetores-m, indicando que o processo das sequências de granizo segue o teorema do limite central. Por fim, é possível conceber o Mapa de Collatz como um destruidor de estruturas, criando e reproduzindo padrões aleatórios, e lentamente destruindo toda e qualquer ordem imposta previamente.Abstract: Arithmetic Dynamics is an emergent research area that studies the behavior of systems in discrete spaces and times. The present dissertation deals with an arithmetic dynamical system called Collatz Map, a rule for positive integers n, stating that n -> n/2 for n even, and n -> 3n + 1 for n odd. A renowned and unsolved mathematical problem conjectures that for any positive integer n, finite iterations of the Collatz Map eventually reach 1. The sequence of integers from iterating the Collatz Map from an initial condition n0 until reach 1 is often called hailstone sequence. The Collatz Map, besides providing very rich dynamics for natural numbers, only recently has been explored in the context of physical models and phenomena. This work describes investigations trying to characterize whether the hailstone sequences can be regarded as a deterministic system performing stochastic-like behavior, aiming to enlighten the path from number theory to statistical mechanics, in the area of stochastic arithmetic dynamical systems. In order to do that, proper statistical analysis of various sequences were done, utilizing a set of very large initial conditions (up to n0 ~ 2^10000). The sampling of initial conditions was conduced by using a new representation for positive integers with direct connection with 2-adics, called m-vectors. By employing methods of time series analysis such as Power Spectrum and Detrended Fluctuation Analysis, the stochastic-like behavior is confirmed, reinforcing literature about the hailstone sequences performing Geometric Brownian Motion (GBM). Autocorrelation function and von Neumann entropy analysis shows deviations from GBM for special initial conditions, indicating sources of determinism and predictability inside the general trend. These deviations appears in the form of short- to mid-range autocorrelations and smaller entropy values for the orbits from these specific initial conditions. The von Neumann entropy also allows the characterization of the internal structure of the sequence by the m-vectors components analysis, indicating that the hailstone sequence is process following a central limit theorem. Finally, it is possible to conceive the Collatz Map as a structure destroyer, that creates and reproduces random patterns, slowly destructing any previously imposed ordering

Weak ergodicity breaking and quantum scars in constrained quantum systems

The success of statistical mechanics in describing complex quantum systems rests upon typicality properties such as ergodicity. Both integrable systems and the recently discovered many-body localisation show that these assumptions can be strongly violated in either finely tuned cases, or in the presence of quenched disorder. In this thesis, we uncover a qualitatively different form of ergodicity breaking, wherein a small number of atypical eigenstates are embedded throughout an otherwise thermalising spectrum. We call this a many-body quantum scar, in analogy to quantum scars in single-particle quantum chaos, where quantum scarred eigenfunctions concentrate around associated periodic classical trajectories. We demonstrate that many-body quantum scars can be found in an unusual model recently realised in a 51 Rydberg atom quantum simulator. The observed coherent oscillations following in a certain quench experiment are a consequence of the quantum scar. At the same time, the level statistics rules out conventional explanations such as integrability and many-body localisation. We develop an approximate method to construct scarred eigenstates, in order to describe their structure and physical properties. Additionally, we find a local perturbation which makes these non-equilibrium properties much more pronounced, with near perfect quantum revivals. At the same time the other eigenstates remain thermal. Our results suggest that many-body quantum scars forms a new class of quantum dynamics with unusual properties, which are realisable in current experiments

Artin's primitive root conjecture -a survey -

This is an expanded version of a write-up of a talk given in the fall of 2000 in Oberwolfach. A large part of it is intended to be understandable by non-number theorists with a mathematical background. The talk covered some of the history, results and ideas connected with Artin's celebrated primitive root conjecture dating from 1927. In the update several new results established after 2000 are also discussed.Comment: 87 pages, 512 references, to appear in Integer

The Turbulent Transport and Biological Structure of Eutrophication Models. Volume I: Preserving the Statistical Structure in Lake Transport Calculations

Project Completion Report, Volume I Office of Water Resources Research and Technology Matching Grant B-O36-OHIO(print) v. ; ill. ; 28 cm.Preface -- Acknowledgments -- List of Tables -- List of Figures -- List of Symbols -- Chapter I. Introduction and Objective -- Chapter II. Review of Turbulent Transport Models -- Chapter III. Turbulence -- Chapter IV. Filtration -- Chapter V. Derivation of Filtered Transport Equations -- Chapter VI. Numerical Solution -- Chapter VII. Model Implementation -- Chapter VIII. Results -- Chapter IX. Interpretation and Discussion -- Chapter X. Conclusions -- Appendix -- Reference